Problem: Find \[\left|\left(1 + \sqrt{3}i\right)^4\right|\]
First of all, we know that $|ab|=|a|\cdot |b|$, so \[\left|\left(1 + \sqrt{3}i\right)^4\right|=\left|1 + \sqrt{3} i\right|^4\]We also find that \[\left|1 +\sqrt{3}i\right|=\sqrt{\left(1\right)^2+\left(\sqrt{3}\right)^2}=\sqrt{4}=2\]Therefore, our answer is $2^4=\boxed{16}$.